Evaluate the following limits :
`Lim_(theta to pi/2) (1-sin theta)/((pi/2-theta)cos theta)`

Answer

To evaluate the limit

\[ L = \lim_{\theta \to \frac{\pi}{2}} \frac{1 - \sin \theta}{\left(\frac{\pi}{2} - \theta\right) \cos \theta} \]

we will follow these steps:

### Step 1: Substitute the limit
First, we substitute \(\theta = \frac{\pi}{2}\) into the expression:

\[ L = \frac{1 - \sin\left(\frac{\pi}{2}\right)}{\left(\frac{\pi}{2} - \frac{\pi}{2}\right) \cos\left(\frac{\pi}{2}\right)} = \frac{1 - 1}{0 \cdot 0} = \frac{0}{0} \]

Since we get an indeterminate form \( \frac{0}{0} \), we can apply L'Hôpital's Rule.

### Step 2: Apply L'Hôpital's Rule
L'Hôpital's Rule states that if we have an indeterminate form \( \frac{0}{0} \), we can differentiate the numerator and the denominator:

\[ L = \lim_{\theta \to \frac{\pi}{2}} \frac{\frac{d}{d\theta}(1 - \sin \theta)}{\frac{d}{d\theta}\left((\frac{\pi}{2} - \theta) \cos \theta\right)} \]

### Step 3: Differentiate the numerator and denominator
The derivative of the numerator \(1 - \sin \theta\) is:

\[ \frac{d}{d\theta}(1 - \sin \theta) = -\cos \theta \]

For the denominator, we will use the product rule:

\[ \frac{d}{d\theta}\left((\frac{\pi}{2} - \theta) \cos \theta\right) = \frac{d}{d\theta}(\frac{\pi}{2} - \theta) \cdot \cos \theta + (\frac{\pi}{2} - \theta) \cdot \frac{d}{d\theta}(\cos \theta) \]

Calculating this gives:

\[ = -\cos \theta + (\frac{\pi}{2} - \theta)(-\sin \theta) = -\cos \theta - (\frac{\pi}{2} - \theta) \sin \theta \]

### Step 4: Substitute again
Now we substitute back into the limit:

\[ L = \lim_{\theta \to \frac{\pi}{2}} \frac{-\cos \theta}{-\cos \theta - (\frac{\pi}{2} - \theta) \sin \theta} \]

### Step 5: Evaluate the limit
Substituting \(\theta = \frac{\pi}{2}\):

\[ L = \frac{-\cos\left(\frac{\pi}{2}\right)}{-\cos\left(\frac{\pi}{2}\right) - \left(\frac{\pi}{2} - \frac{\pi}{2}\right) \sin\left(\frac{\pi}{2}\right)} = \frac{0}{0 - 0} = \frac{0}{0} \]

Since we still have \( \frac{0}{0} \), we apply L'Hôpital's Rule again.

### Step 6: Differentiate again
Differentiating again gives:

Numerator:

\[ \frac{d}{d\theta}(-\cos \theta) = \sin \theta \]

Denominator:

\[ \frac{d}{d\theta}(-\cos \theta - (\frac{\pi}{2} - \theta) \sin \theta) = \sin \theta - \left(-\sin \theta + (\frac{\pi}{2} - \theta) \cos \theta\right) = 2\sin \theta - (\frac{\pi}{2} - \theta) \cos \theta \]

### Step 7: Substitute again
Now substituting again:

\[ L = \lim_{\theta \to \frac{\pi}{2}} \frac{\sin \theta}{2\sin \theta - (\frac{\pi}{2} - \theta) \cos \theta} \]

### Step 8: Final evaluation
Substituting \(\theta = \frac{\pi}{2}\):

\[ L = \frac{\sin\left(\frac{\pi}{2}\right)}{2\sin\left(\frac{\pi}{2}\right) - 0} = \frac{1}{2 \cdot 1} = \frac{1}{2} \]

Thus, the limit evaluates to:

\[ \boxed{1} \]